The QUANTREG Procedure

Quantile Regression as an Optimization Problem

The generic model for linear quantile regression is

upper Q Subscript tau Baseline left-parenthesis upper Y vertical-bar upper X equals bold x right-parenthesis equals upper Q Subscript upper Y vertical-bar bold x Baseline left-parenthesis tau right-parenthesis equals bold x prime bold-italic beta left-parenthesis tau right-parenthesis

where Y is the response random variable, is the explanatory covariates vector, is the vector of the functional model parameters at the quantile level , and is the quantile function for Y conditional on .

This generic model is compatible with the following1 linear model:

y Subscript i Baseline equals bold x prime Subscript i Baseline bold-italic beta left-parenthesis tau right-parenthesis plus epsilon Subscript i Baseline left-parenthesis tau right-parenthesis for i equals 1 comma ellipsis comma n

where is the response value, is the explanatory covariates vector, and is an unknown error.

regression, also known as median regression, is a natural extension of the sample median when the response is conditioned on the covariates. In regression, the least absolute residuals estimate , referred to as the -norm estimate, is obtained as the solution of the following minimization problem:

min Underscript bold-italic beta element-of bold upper R Superscript p Endscripts sigma-summation Underscript i equals 1 Overscript n Endscripts StartAbsoluteValue y Subscript i Baseline minus bold x prime Subscript i Baseline bold-italic beta EndAbsoluteValue

More generally, for quantile regression Koenker and Bassett (1978) defined the regression quantile, , as any solution to the following minimization problem:

min Underscript bold-italic beta element-of bold upper R Superscript p Endscripts left-bracket sigma-summation Underscript i element-of StartSet i colon y Subscript i Baseline greater-than-or-equal-to bold x prime Subscript i Baseline bold-italic beta EndSet Endscripts tau StartAbsoluteValue y Subscript i Baseline minus bold x prime Subscript i Baseline bold-italic beta EndAbsoluteValue plus sigma-summation Underscript i element-of StartSet i colon y Subscript i Baseline less-than bold x prime Subscript i Baseline bold-italic beta EndSet Endscripts left-parenthesis 1 minus tau right-parenthesis StartAbsoluteValue y Subscript i Baseline minus bold x prime Subscript i Baseline bold-italic beta EndAbsoluteValue right-bracket

The solution is denoted as , and the -norm estimate corresponds to . The regression quantile is an extension of the sample quantile , which can be formulated as the solution of

min Underscript xi element-of bold upper R Endscripts left-bracket sigma-summation Underscript i element-of StartSet i colon y Subscript i Baseline greater-than-or-equal-to xi EndSet Endscripts tau StartAbsoluteValue y Subscript i Baseline minus xi EndAbsoluteValue plus sigma-summation Underscript i element-of StartSet i colon y Subscript i Baseline less-than xi EndSet Endscripts left-parenthesis 1 minus tau right-parenthesis StartAbsoluteValue y Subscript i Baseline minus xi EndAbsoluteValue right-bracket

If you specify weights , with the WEIGHT statement, weighted quantile regression is carried out by solving

min Underscript bold-italic beta Subscript w Baseline element-of bold upper R Superscript p Endscripts left-bracket sigma-summation Underscript i element-of StartSet i colon y Subscript i Baseline greater-than-or-equal-to bold x prime Subscript i Baseline bold-italic beta Subscript w Baseline EndSet Endscripts w Subscript i Baseline tau StartAbsoluteValue y Subscript i Baseline minus bold x prime Subscript i Baseline bold-italic beta Subscript w Baseline EndAbsoluteValue plus sigma-summation Underscript i element-of StartSet i colon y Subscript i Baseline less-than bold x prime Subscript i Baseline bold-italic beta Subscript w Baseline EndSet Endscripts w Subscript i Baseline left-parenthesis 1 minus tau right-parenthesis StartAbsoluteValue y Subscript i Baseline minus bold x prime Subscript i Baseline bold-italic beta Subscript w Baseline EndAbsoluteValue right-bracket

Weighted regression quantiles can be used for L-estimation (Koenker and Zhao 1994).

Last updated: December 09, 2022